In this talk, we explore explicit cross sections to the horocycle and geodesic flows on $\operatorname{SL}(2, \mathbb{R})/G_q$, with $q \geq 3$. Our approach relies on extending properties of the primitive integers $\mathbb{Z}_\text{prim}^2 := \{(a, b) \in \mathbb{Z}^2 \mid \gcd(a, b) = 1\}$ to the discrete orbits $\Lambda_q := G_q (1, 0)^T$ of the linear action of $G_q$ on the plane $\mathbb{R}^2$. We present an algorithm for generating the elements of $\Lambda_q$ that extends the classical Stern-Brocot process, and from that derive another algorithm for generating the elements of $\Lambda_q$ in planar strips in increasing order of slope. We parametrize those two algorithm using what we refer to as the \emph{symmetric $G_q$-Farey map}, and \emph{$G_q$-BCZ map}, and demonstrate that they are the first return maps of the geodesic and horocycle flows resp. on $\operatorname{SL}(2, \mathbb{R})/G_q$ to particular cross sections. Using homogeneous dynamics, we then show how to extend several classical results on the statistics of the Farey fractions, and the symbolic dynamics of the geodesic flow on the modular surface to our setting using the $G_q$-BCZ and symmetric $G_q$-Farey maps. This talk is self-contained, and does not assume any prior knowledge of Hecke triangle groups or homogeneous dynamics.